Properties

Label 2.12.12.26
Base \(\Q_{2}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(12\)
Galois group $C_6\times C_2$

Related objects

Learn more about

Defining polynomial

\( x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $6$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{2}$
Root number: $-i$
$|\Gal(K/\Q_{ 2 })|$: $12$
This field is Galois and abelian over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-*})$, 2.3.0.1, 2.4.4.1, 2.6.0.1, 2.6.6.3, 2.6.6.5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{6} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} - 2 x + 8 t^{2} + 8 t + 6 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_6\times C_2$
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:[2]
Galois mean slope:$1$
Global splitting model:\( x^{12} - x^{6} + 1 \)